Theorem xmstopn 24659

Description: The topology component of an extended metric space coincides with the topology generated by the metric component. (Contributed by Mario Carneiro, 26-Aug-2015.)

Hypotheses

Ref	Expression
isms.j	⊢ 𝐽 = (TopOpen‘𝐾)
isms.x	⊢ 𝑋 = (Base‘𝐾)
isms.d	⊢ 𝐷 = ((dist‘𝐾) ↾ (𝑋 × 𝑋))

Assertion

Ref	Expression
xmstopn	⊢ (𝐾 ∈ ∞MetSp → 𝐽 = (MetOpen‘𝐷))

Proof of Theorem xmstopn

Step	Hyp	Ref	Expression
1		isms.j	. . 3 ⊢ 𝐽 = (TopOpen‘𝐾)
2		isms.x	. . 3 ⊢ 𝑋 = (Base‘𝐾)
3		isms.d	. . 3 ⊢ 𝐷 = ((dist‘𝐾) ↾ (𝑋 × 𝑋))
4	1, 2, 3	isxms 24655	. 2 ⊢ (𝐾 ∈ ∞MetSp ↔ (𝐾 ∈ TopSp ∧ 𝐽 = (MetOpen‘𝐷)))
5	4	simprbi 503	1 ⊢ (𝐾 ∈ ∞MetSp → 𝐽 = (MetOpen‘𝐷))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2146   × cxp 5661   ↾ cres 5665  ‘cfv 6540  Basecbs 17291  distcds 17341  TopOpenctopn 17496  MetOpencmopn 21562  TopSpctps 23139  ∞MetSpcxms 24525

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-xp 5669  df-res 5675  df-iota 6496  df-fv 6548  df-xms 24528

This theorem is used by:  imasf1oxms  24697  ressxms  24733  prdsxmslem2  24737  tmsxpsmopn  24745  xmsusp  24777  cmetcusp1  25563  minveclem4a  25640  minveclem4  25642  qqhcn  34445  rrhcn  34451  rrexthaus  34461  dya2icoseg2  34733  sitmcl  34806